Stochastic Thermodynamics

Transcript

1 Leiden, Hot nanostructures, October 213 Stochastic Thermodynamics Udo Seifert II. Institut für Theoretische Physik, Universität Stuttgart Review: U.S., Rep. Prog. Phys ,

2 Thermodynamics of macroscopic systems [19 th cent] λ W T λ t First law energy balance: W = E +Q = E +T S M Second law: S tot S + S M > W > E T S F W diss W F > 2

3 Stochastic thermodynamics for small systems W λ T, p λ t driving: mechanical (electro)chemical (bio)chemical First law: how to define work, internal energy and exchanged heat? fluctuations imply distributions: p(w; λ(τ))... entropy: distribution as well? 3

4 Stochastic thermodynamics applies to such systems where non-equilibrium is caused by mechanical or chemical forces ambient solution provides a thermal bath of well-defined T and µ i fluctuations are relevant due to small numbers of involved molecules Main idea: Energy conservation (1 st law) and entropy production (2 nd law) along an individual stochastic trajectory Precursors: notion stoch th dyn by Nicolis, van den Broeck mid 8s ( ensemble level) early fluct diss response relations: Hänggi & Thomas early 8 s stochastic energetics (1 st law) by Sekimoto late 9s work theorem(s): Jarzynski, Crooks late 9s, Imparato & Peliti early s fluct theorem: Evans, Cohen, Galavotti, Kurchan, Lebowitz & Spohn 9s quantities like stochastic entropy by Crooks, Qian, Gaspard in early s... 4

5 Paradigm for mechanical driving: x x 4 x 1 x 6 x 3 x 5 x 2 λ(τ) V (x, λ) λ(τ) x f(λ) V (x,λ) Langevin dyn s ẋ = µ[ V (x,λ)+f(λ)] }{{} F(x,λ) external protocol λ(τ) +ζ ζζ = 2µk B T δ(...) }{{} ( 1) First law [(Sekimoto, 1997)]: dw = du+dq applied work: dw = λ V(x,λ)dλ+f(λ)dx internal energy: du = dv dissipated heat: dq = dw du = F(x,λ)dx = Tds m 5

6 Experimental illustration: Colloidal particle in V(x, λ(τ)) [V. Blickle, T. Speck, L. Helden, U.S., C. Bechinger, PRL 96, 763, 26] work distribution non-gaussian distribution Langevin valid beyond lin response [T. Speck and U.S., PRE 7, 66112, 24] 6

7 Stochastic entropy [U.S., PRL 95, 462, 25] Fokker-Planck equation τ p(x,τ) = x j(x,τ) = x (µf(x,λ) D x )p(x,τ) [D = µk B T] Common non-eq ensemble entropy [k B 1] S(τ) dx p(x,τ)lnp(x,τ) Stochastic entropy for an individual trajectory x(τ) s(τ) lnp(x(τ),τ) with s(τ) = S(τ) s tot s m + s exp[ s tot ] = 1 s tot integral fluctuation theorem for total entropy production arbitrary initial state, driving, length of trajectory 7

8 Heat engines at maximal power Carnot (1824) Curzon-Ahlborn (1975) Q h T h Q h = α(t h T m h ) T m h T h W W T m c Qc η c 1 T c /T h but zero power Tc Qc = β(t m c Tc) efficiency at maximum power η ca 1 T c /T h Tc 8

9 Brownian heat engine at maximal power with universal bounds [T. Schmiedl and U.S., EPL 81, 23, (28)] V 1 V p a p b 1 V 4 T h T c 3 V 2 η α = 1/2 α = α CA α = 1 p a p b η C (T h T c )/T c η = η c 2 αη c with α 1 implies η c 2 η η c 2 η c Curzon-Ahlborn neither universal nor a bound cf. Esposito, Kawai, Lindenberg, vd Broeck, PRL 1,... 9

10 Experimental realization in the Stirling version H transition isochoric [V. Blickle and C. Bechinger, Nature phys. 8, 143, 212] 4 2 a W [k B T c ] W [k B T c ] 2 1 (1) (2) (1) (2) (1) (2) w n T =86 C h (3) (4) -8 (3) (4) (3) (4) (3) (4) -1 time[s] w n [k B T c ] 1 b time[s] 1 c position [µm] p isothermal expansion macroscopic Stirling cycle position [µm] cycle number n -1 1 p(w n ).2 (3).. isochoric transition (2) (2) (4) (1) V (1) P [k B T c /s] a w [k B T c ] T =22 C c position [µm] isothermal compression position [µm] Efficiency. b [s] 1

11 NESSs: Examples and common characteristics f(λ) V (x,λ) Time-independent driving beyond linear response regime Broken detailed-balance Persistent currents with permanent dissipation Stationary (non-boltzmann) distribution 11

12 Fluctuation theorem p( s tot )/p( s tot ) = exp( s tot ) long-time limit: Evans et al (1993), Gallavotti & Cohen (1995), Kurchan (1998), Lebowitz & Spohn (1999)... finite times: U.S., PRL 5 experimental data [Speck, Blickle, Bechinger, U.S., EPL (27)] ~ a ) t = 2 s t = 2 s f(λ) V (x,λ) b ) t = 2 s -25 t = 2 s t = ẋ = µ[ x V(x)+f]+ζ, to ta l e n tr o p y p r o d u c tio n s t o t [k B ] 12

13 F theorem and slow hidden degrees of freedom [J. Mehl, B. Lander, C. Bechinger, V. Blickle and U.S., PRL 18, 2261, 212] total entropy production in the NESS s tot t dτ[ x 1 ν 1 (x 1,x 2 )+ x 2 ν 2 (x 1,x 2 )] with ν 1 (x 1,x 2 ) x 1 x 1,x 2 x 1 x 2 (a) + - R f(λ) R f 1 V (x,λ) f(λ) V (x,λ) min.,15,165,18,195,21,225,24,255,27,285,3 max. +1 x ( R) obeys FT p( s tot )/p( s tot ) = exp s tot B x 1 (b) suppose x 2 is hidden: ν 1 (x 1 ) ν(x 1,x 2 )p(x 2 x 1 )dx 2 apparent entropy production s tot t dτx 1 ν 1 (x 1 ) obeys FT?? + - R potential (units of k B T) x U 1 R f 2 U x 1, x 2 ( R) (c) (d) x 1 ( R) x ( R) x ( R) 13

14 ~ ~ Experimental data with and without coupling [rarely:] p(ds tot )(1-2 ) ~ ~ Ds tot ln[p(ds tot )/p(-ds tot )] (c) ~ Ds tot ~ ~ ln[p(ds tot )/p(-ds tot )] ~ Ds tot (a) slope α 1,1 1, slope a,9,8,7 (a), G,5 1, 1,5 2, t(s) 14

15 Fluctuation-dissipation (response) theorem (FDT) in equilibrium perturbation with a field f : E E fb observable A T δa(t 2) δf(t 1 ) = t 1 A(t 2 )B(t 1 ) any variable A formalizes Onsager s regression hypothesis in a NESS often (phenomenologically) modified by an effective temp: T eff δa(t 2 ) δf(t 1 ) = t 1 A(t 2 )B(t 1 ) Culgiandolo, Kurchan & Peliti, 97 Berthier and Barrat,

16 Paradigm for an FDT in a NESS [T. Speck and U.S., Europhys. Lett. 74, 391, 26] f(λ) V (x,λ) FDT in eq: T δ ẋ(t 2) δf(t 1 ) f= = ẋ(t 2 )ẋ(t 1 ) eq extended FDT in non-eq: T δ ẋ(t 2) δf(t 1 ) f = ẋ(t 2 )ẋ(t 1 ) ness ẋ(t 2 )ν s (x(t 1 )) ness with ν s (x) ẋ x = j s /p s (x) additive modification (rather than multiplicative) 16

17 Integrated version: Generalized Einstein relation [Blickle, Speck, Lutz, U.S., Bechinger, PRL 98, 2161 (27)] f(λ) V (x,λ) δ ẋ(t 2 ) = ẋ(t 2 )ẋ(t 1 ) neq ẋ(t 2 )ν s (x(t 1 )) neq δf(t 1 ) f µ eff (f) = D eff (f) dt I(t) with I(t) ẋ(t)ν s(x()) ν s (x) v io la tio n in te g r a l m o b ility D cf giant diffusion [Reimann et al 21]

18 General FDT in a NESS [U.S and T. Speck, EPL 89: 17, 21] NESS : δa(t 2 ) δf(t 1 ) = A(t 2 ) f ṡ(t 1 ) = A(t 2 ) f ṡ m (t 1 ) }{{} ness + A(t 2 ) f ṡ tot (t 1 ) ness EQ : T δa(t 2) δf(t 1 ) = A(t 2 ) f Ė(t 1 ) eq stochastic entropy replaces energy add to eq form the term conj to total entropy production proof: (1) pert theory of FPE + (2) use of det bal in equilibrium 18

19 Sheared colloidal suspension with 3d Brownian dynamics [B. Lander, U.S. and T. Speck, EPL , 21; PRE 85, 2113, 212] driving: linear shear flow perturbating field: force on tagged particle 19

20 velocity-force FDT in a NESS 1.5 γ = 1., φ =.1 γ = 1., φ =.3 R ij (t) = δ v i(t) δf j () 1 C xx R xx C xx R xx C ij (t) = v i (t)v j () t t γ = 1., φ =.1 γ = 1., φ =.3 effective temperature? θ i C ii(t) R ii (t) 1+a γ2 works well at larger densities θ x θ y θ z γ θ x θ y θ z γ t t 2

21 effective confinement for moderate/large densities Cxx,Rxx m=1,k=1.25,θ x =2.6 A mc xx /θ x mr xx t/m effective temperature also for trapped particle? similar math. structure in confined suspensions C ii (t) θ i R ii (t)+ γ 2 β i I i (t) with max I i (t) = 1 βx Cxx,Rxx Cxx,Rxx 1. m=1,k=12.5,θ x =1.16 B t/m 1. m=.5,k=25,θ x =1.4 C t/m 3 D E 2 X x θ x k m=1 1 m=.1 θ x k γ 21

22 An isothermal nano-rotor: F1-ATP-ase [K. Hayashi,... H. Noji, PRL 14, (21)] kinetics vs thermodynamics first law? efficiency(ies)? 22

23 Hybrid model [E. Zimmermann and U.S., New J. Phys. 14, 1323, 212] probe particle ẋ = µ( y V(y)+f ex )+ζ with y(τ) n(τ) x(τ) motor w + /w = exp[ µ V(n+d,x) V(n,x)] local detailed balance condition 23

24 Simulated trajectories t [s] c ATP = M x [12 ] c ATP = M c ATP = M probe motor 24

25 F1-ATPase and the fluctuation theorem [K. Hayashi,... H. Noji, PRL 14, (21)] Γ θ = N +ζ ζ 1 ζ 2 ) = 2Γk B Tδ(τ 1 τ 2 ) ln[p( θ)/p( θ)] = N θ/k B T independent of friction coefficient cf f theorem for total entropy production time-dependence? ln[p( s tot )/p( s tot )] = s tot /k B 25

26 FT-slope from simulations vs experiment p( x) x [d] t = 1 ms t = 2 ms t = 5 ms t = 1 ms t = 2 ms c ATP = c ATP = c ATP = c ATP = α t 26

27 Efficiencies Thermodynamic efficiency [Parmeggiani et al, PRE 1999] η f ex v/ µ for f ex = : pseudo efficiency [Toyabe et al, PRL 21] η Q Q p / µ 27

28 Experimental determination of η Q [S. Toyabe et al, PRL 14, (21)] Harada-Sasa relation [PRL 26] µ Q P = v 2 + dω[cẋ(ω) 2k B T ReRẋ(ω)] from violation of fluc-diss-theorem 28

29 Comparison to experiment quite reasonable agreement for small θ + future: substeps of degrees 29

30 Efficiency of autonomous nano-machines at maximum power [U.S., PRL 16, 261, 211] output: w out = fd input: w in = µ mean currents: j ± local detailed balance: j /j + = exp( S) = exp(w out w in ) power: P out,in = (j + j ) w out,in = j + (w out,w in ) [1 exp(w in w out )] w out,in efficiency: η P out /P in (= w out /w in, for unicyclic) 3

31 Efficiency at maximum power (EMP) case I: fix w in ( µ), maximize P out with respect to w out ( f): EMP: η w out w in 1/2+(x out +1/2)w in. with x out dlnj + /dw out 1/2 universal in LR for strong coupling (Kedem and Caplan, 1965; vd Broeck and Esposito 25...) 1. x out 5 Η x out 1 2 x out Ω in 31

32 Efficiency at maximum power η case III: maximize P out with respect to w out ( f) and w in ( µ) three regimes (determined by two structural parameters) 32

33 Thermoelectrics [Snyder et al, Nat. Mat. 7 15, 28] 33

34 Paradigm for thermoelectrics: Single level quantum dot heat engine [Linke et al 22..., Esposito, vd Broeck et al 25...] particle current j = (w + 1 w+ 2 w 1 w 2 )/Σw th dynamic consistency: w + 1 /w 1 = exp(µ 1 E))/T 1, w + 2 /w 2 =... output power P out = (µ 2 µ 1 )j input power (heat from hot reservoir) P in = (E µ 1 )j efficiency η P out /P in = (µ 2 µ 1 )/(E µ 1 ) η C maximize P out wrt (µ 2 µ 1 ) at fixed T 2 T 1,E : η η C /2+O(η 2 C )... and with respect to E : η η C /2+η 2 C /8 34

35 Thermoelectrics with magnetic field μ L T L B C μ R T R speculation: Carnot efficiency at finite power? Benenti et al, PRL 211 linear response regime ( Jρ J q ) = ( Lρρ L ρq L qρ L qq )( Fρ F q ) fundamental constraints 2 nd law Onsager relations 4L ρρ L qq (L ρq +L qρ ) 2 L ρρ,l qq L ρq (B) = L qρ ( B) 35

36 Three-Terminal Model [K. Brandner, K. Saito, U.S., PRL , 213] effective Onsager matrix ( Lρρ L ρq L qρ L qq ) = L LL L LP L 1 PP L PL μ L T L μ P B T P C μ R T R multiterminal Landauer formula L AB = F(E) [ new bound de F(E) ( (E µ)e (E µ)e (E µ) 2 e 2 4k B hcosh 2 ( E µ k B T )] 1 ) (δab S AB(E,B) 2) 4L ρρ L qq (L ρq +L qρ ) 2 3(L ρq L qρ ) 2 36

37 Efficiency dimensionless parameters y L ρq L qρ /(L ρρ L qq L ρq L qρ ) x L ρq /L qρ η max/ηc bound on η max η max η C = bound on η(p max ) η (P max ) η C = x 2 x+1 x 1 x 2 x+1 x 1 x 2 4x 2 6x+4 η (Pmax)/ηC x x 37

38 Multi-terminal model [K. Brandner, U.S., NJP , 213] 1 μ 1 T 1.8 μ3 T3 μ 2 T 2 B μn T n ηmax(x)/ηc x μ4 T4.8 generalized bound 4L ρρ L qq (L ρq +L qρ ) 2 tan ( ) π n (L ρq L qρ ) 2 η (x)/ηc x 38

39 A classical Nernst engine [J. Stark, K. Brandner, K. Saito, U.S., arxiv: ] 5 B -βµ 2 5 A figure of merit ZT = L 2 ρq L ρρ L qq L 2 ρq = (NB)2 σt κ 2 nd law ZT 1 new bound ZT 1/2 η max η C = 1 1 ZT 1+ 1 ZT η(p max ) η C = ZT 4 2ZT 1/6 39

40 Summary and Perspectives Stochastic thermodynamics along individual trajectories formulation of the 1 st law refinement of the 2 nd law NESS extended FDT ( Green-Kubo for a NESS) emergence of an effective temperature in sheared suspensions Efficiency thermolectric transport molecular motors information machines (in cell biology) 4